TITLE: Semigroup analogue to the classification of finite simple groups?
QUESTION [11 upvotes]: Is there a semigroup analogue to the classification of finite simple groups?
If so what are some of the major results?
--Edit, reference links---
Classification of Finite Simple Groups
Special Classes of Semigroups 
One of my semigroup equation sequences in OEIS, g(f(x)) = f(f(f(x)))

REPLY [5 votes]: In semigroup theory, the term "simple" has a specialized meaning.  If you interpret it to mean that the semigroup has no quotients other than itself and the trivial semigroup, then there is a complete classification.  In semigroup theory, these are called "congruence-free semigroups".  (In universal algebra, "simple" means no non-trivial quotients.)
There is a complete classification of finite congruence-free semigroups.  They are either:

a two-element semigroup
a finite simple group
a completely 0-simple semigroup.  Most of these are not congruence-free, but the ones that are have a complete description.  They are basically paramatrized by 0-1 matrices.

The proof that a congruence-free semigroup that is not covered by cases 1 or 2 must have a zero is sketched in this Math Overflow answer.  (A zero is just an element $0$ such that $0a = a0 = 0$ for all $a$.)
I don't know of a great online source for the third case.  This preprint states the theorem, with reference, as Theorem 3.1, but it's not the main subject of the paper.